The book explores the theory of asymptotic martingales using a measure-theoretic framework, providing a comprehensive introduction to this advanced statistical concept. It delves into the properties and applications of asymptotic martingales, offering insights into their behavior and significance in probability theory. Additionally, the text includes a bibliography that serves as a valuable resource for further reading and research in the field, making it a useful reference for both students and professionals interested in advanced probability topics.
Twenty-five years ago, Hans Blihlmann's influential monograph established nonlife actuarial mathematics as a recognized field within probability theory and statistics, with connections to economics. This work served as my guide when I taught my first course on nonlife actuarial mathematics in Summer 1988, while also integrating insights from the rapidly expanding literature inspired by Blihlmann's contributions. The current book focuses solely on the temporal development of a fixed portfolio of risks, emphasizing the claim number process and its related concepts, including the claim arrival process, aggregate claims process, risk process, and reserve process. It highlights characterizations of various claim number process classes, offering alternative criteria for model selection, and explores their relationships with the binomial, Poisson, and negative-binomial distributions. The mixed Poisson process is given special attention due to its applicability in numerous contexts. Additionally, the book addresses important issues such as thinning, decomposition, and superposition of risk processes, which are crucial for reinsurance considerations, as well as the role of martingales that naturally arise in canonical scenarios.
